Polytope of Type {4,12} with group SmallGroup(1920,240505)

Overview

Group: SmallGroup(1920,240505)
Rank: 3
Schläfli Type: {4,12}
Vertices, edges, …: 80, 480, 240
Order of s₀s₁s₂: 20
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

{4,6}*120

60-fold

{4,4}*32

120-fold

240-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1*s2*s1)^3> of order 2

120 facets

120 of {4}*8

40 vertex figures

40 of {12}*24

P/N, where N=<s1*s0*s1*s2*s1*s0*(s2*s1)^4*s0*s2*s1*s2> of order 2

120 facets

120 of {4}*8

40 vertex figures

40 of {12}*24

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^3*s0*s1*s2> of order 2

120 facets

120 of {4}*8

40 vertex figures

40 of {12}*24

P/N, where N=<s0*s2*s1*s0*(s1*s2)^2*(s1*s0*s2)^2*s1*s2> of order 2

120 facets

120 of {4}*8

40 vertex figures

40 of {12}*24

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2> of order 2

120 facets

120 of {4}*8

40 vertex figures

40 of {12}*24

P/N, where N=<s0*(s2*s1)^2*s0*s2*s1*s0*(s1*s2)^2*s1*s0*s2> of order 2

128 facets

112 of {4}*8
16 of {2}*4

40 vertex figures

40 of {12}*24

P/N, where N=<s1*s0*s1*(s2*s1*s0)^2*(s1*s2)^2*s1*s0*s1*s2> of order 2

120 facets

120 of {4}*8

40 vertex figures

40 of {12}*24

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1*s0> of order 3

80 facets

80 of {4}*8

32 vertex figures

24 of {12}*24
8 of {4}*8

P/N, where N=<s0*s2*s1*s0*(s1*s2)^2*(s1*s0*s2)^2*s1*s2, (s0*s2*s1)^2*s0*(s2*s1)^3*s0*s1*s2> of order 4

60 facets

60 of {4}*8

20 vertex figures

20 of {12}*24

P/N, where N=<s0*s2*s1*s0*(s1*s2)^2*(s1*s0*s2)^2*s1*s2, (s0*s2*s1)^2*s0*(s1*s2)^2*s1*s0*s2*s1*s2> of order 4

60 facets

60 of {4}*8

20 vertex figures

20 of {12}*24

P/N, where N=<(s0*s1)^2, (s2*s1)^2*s0*s2*s1*s0*(s1*s2)^3> of order 4

68 facets

16 of {2}*4
52 of {4}*8

20 vertex figures

20 of {12}*24

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2, s1*s0*s1*(s2*s1*s0)^2*(s1*s2)^2*s1*s0*s1*s2> of order 4

60 facets

60 of {4}*8

20 vertex figures

20 of {12}*24

P/N, where N=<s0*s2*s1*s0*(s1*s2)^2*(s1*s0*s2)^2*s1*s2, (s0*s2*s1)^2*s0*(s2*s1)^3*s0*s2*s1*s2> of order 4

60 facets

60 of {4}*8

20 vertex figures

20 of {12}*24

P/N, where N=<s0*s2*s1*s0*(s1*s2)^2*(s1*s0*s2)^2*s1*s2, s1*s0*s1*(s2*s1*s0)^2*(s1*s2)^2*s1*s0*s1*s2> of order 4

60 facets

60 of {4}*8

20 vertex figures

20 of {12}*24

P/N, where N=<s0*(s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1> of order 5

48 facets

48 of {4}*8

16 vertex figures

16 of {12}*24

P/N, where N=<(s0*(s2*s1)^2)^2, (s0*s1)^2*(s2*s1)^2*(s0*s2*s1)^2> of order 6

40 facets

40 of {4}*8

16 vertex figures

12 of {12}*24
4 of {4}*8

P/N, where N=<s0*(s1*s2)^3*s1*s0*s1*s2*s1, (s1*s2)^2*(s1*s0*s1*s2)^2> of order 6

40 facets

40 of {4}*8

16 vertex figures

12 of {12}*24
4 of {4}*8

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1*s0> of order 6

48 facets

32 of {4}*8
16 of {2}*4

16 vertex figures

12 of {12}*24
4 of {4}*8

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1*s0, s1*s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*(s1*s2)^2> of order 6

40 facets

40 of {4}*8

16 vertex figures

12 of {12}*24
4 of {4}*8

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1*s0, (s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1*s0*s2*s1> of order 6

40 facets

40 of {4}*8

16 vertex figures

12 of {12}*24
4 of {4}*8

P/N, where N=<(s0*s1)^2, (s2*s1)^2*s0*s2*s1*s0*(s1*s2)^3, s0*s1*(s2*s1*s0)^2*(s1*s2)^2*s1*s0*s1*s2*s1> of order 8

34 facets

8 of {2}*4
26 of {4}*8

10 vertex figures

10 of {12}*24

P/N, where N=<(s0*s2*s1)^4, (s0*s1)^2*s2*s1*s0*(s2*s1)^3*s0> of order 10

24 facets

24 of {4}*8

8 vertex figures

8 of {12}*24

P/N, where N=<s0*(s1*s2)^3*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*(s1*s2)^3> of order 10

24 facets

24 of {4}*8

8 vertex figures

8 of {12}*24

P/N, where N=<(s0*s1)^2, (s0*s2*s1)^2*s0*s1*s2*s1*s0*s2> of order 10

32 facets

16 of {2}*4
16 of {4}*8

8 vertex figures

8 of {12}*24

P/N, where N=<(s0*s1)^2*s2*s1*s0*s2*s1*s2> of order 10

24 facets

24 of {4}*8

8 vertex figures

8 of {12}*24

P/N, where N=<s0*(s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1, s0*s1*s0*(s2*s1)^3*s0*(s1*s2)^2> of order 10

24 facets

24 of {4}*8

8 vertex figures

8 of {12}*24

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2, s1*s0*(s2*s1)^2*s0*(s1*s2)^2*s1> of order 12

20 facets

20 of {4}*8

12 vertex figures

4 of {12}*24
8 of {4}*8

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*(s1*s0*s2)^2*s1*s0*s1*s2> of order 20

12 facets

12 of {4}*8

4 vertex figures

4 of {12}*24

Representations

Permutation Representation (GAP)

s0 := ( 2, 4)( 3, 5)(10,13);;
s1 := ( 1, 4)( 2, 7)( 3, 8)( 5, 6)(10,11)(12,13);;
s2 := ( 1, 8)( 6, 7)( 9,11);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(13)!( 2, 4)( 3, 5)(10,13);
s1 := Sym(13)!( 1, 4)( 2, 7)( 3, 8)( 5, 6)(10,11)(12,13);
s2 := Sym(13)!( 1, 8)( 6, 7)( 9,11);
poly := sub<Sym(13)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 >;

References

None.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle